On localizations of quasi-simple groups with given countable center

Abstract

A group homomorphism $i:H\to G$ is a localization of $H$, if for every homomorphism $\phi :H\to G$ there exists a unique endomorphism $\psi :G\to G$ such that $i\psi =\phi$ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e., a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.